---
_id: '65073'
abstract:
- lang: eng
  text: We study the large-time behavior of the continuous-time heat kernel and of
    solutions to the heat equation on homogeneous trees. First, we derive sharp asymptotic
    formulas for the heat kernel as $t\to\infty$. Second, using them, we show that
    solutions with initial data in weighted $\ell^1$ classes, asymptotically factorize
    in $\ell^p$ norms, $p\in[1,\infty]$, as the product of the heat kernel, times
    a $p$-mass function, dependent on the initial condition and $p$. The  $p$-mass
    function is described in terms of boundary averages associated with Busemann functions
    for $p<2$, while for $p\ge 2$, it is expressed through convolution with the ground
    spherical function. For comparison, the case of the integers shows that a single
    constant mass determines the asymptotics of solutions to the heat equation for
    all $p$, emphasizing the influence of the graph geometry on heat diffusion.
author:
- first_name: Efthymia
  full_name: Papageorgiou, Efthymia
  id: '100325'
  last_name: Papageorgiou
citation:
  ama: Papageorgiou E. Long-time asymptotics for the heat kernel and for heat equation
    solutions on homogeneous trees. <i>260311232</i>. Published online 2026.
  apa: Papageorgiou, E. (2026). Long-time asymptotics for the heat kernel and for
    heat equation solutions on homogeneous trees. In <i>2603.11232</i>.
  bibtex: '@article{Papageorgiou_2026, title={Long-time asymptotics for the heat kernel
    and for heat equation solutions on homogeneous trees}, journal={2603.11232}, author={Papageorgiou,
    Efthymia}, year={2026} }'
  chicago: Papageorgiou, Efthymia. “Long-Time Asymptotics for the Heat Kernel and
    for Heat Equation Solutions on Homogeneous Trees.” <i>2603.11232</i>, 2026.
  ieee: E. Papageorgiou, “Long-time asymptotics for the heat kernel and for heat equation
    solutions on homogeneous trees,” <i>2603.11232</i>. 2026.
  mla: Papageorgiou, Efthymia. “Long-Time Asymptotics for the Heat Kernel and for
    Heat Equation Solutions on Homogeneous Trees.” <i>2603.11232</i>, 2026.
  short: E. Papageorgiou, 2603.11232 (2026).
date_created: 2026-03-20T17:55:24Z
date_updated: 2026-03-20T17:55:30Z
external_id:
  arxiv:
  - '2603.11232'
language:
- iso: eng
project:
- _id: '357'
  name: 'TRR 358: Ganzzahlige Strukturen in Geometrie und Darstellungstheorie'
publication: '2603.11232'
status: public
title: Long-time asymptotics for the heat kernel and for heat equation solutions on
  homogeneous trees
type: preprint
user_id: '100325'
year: '2026'
...
---
_id: '64267'
abstract:
- lang: eng
  text: "Let $\\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $Δ$ its
    nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote
    by $λ_{0}$, and $σ\\in (0,1)$.\r\n  The aim of this paper is twofold. On the one
    hand, we determine the Fujita exponent for the fractional heat equation \\[\\partial_{t}
    u + Δ^σu = e^{βt}|u|^{γ-1}u,\\] by proving that nontrivial positive global solutions
    exist if and only if $γ\\geq 1 + β/ λ_{0}^σ$. On the other hand, we prove the
    existence of non-negative, bounded and finite energy solutions of the semilinear
    fractional elliptic equation \\[\r\n  Δ^σ v - λ^σ v - v^γ=0 \\] for $0\\leq λ\\leq
    λ_{0}$ and $1<γ< \\frac{n+2σ}{n-2σ}$. The two problems are known to be connected
    and the latter, aside from its independent interest, is actually instrumental
    to the former.\r\n  \\smallskip\r\n  At the core of our results stands a novel
    fractional Poincaré-type inequality expressed in terms of a new scale of $L^{2}$
    fractional Sobolev spaces, which sharpens those known so far, and which holds
    more generally on Riemannian symmetric spaces of non-compact type. We also establish
    an associated Rellich--Kondrachov-like compact embedding theorem for radial functions,
    along with other related properties."
author:
- first_name: Tommaso
  full_name: Bruno, Tommaso
  last_name: Bruno
- first_name: Effie
  full_name: Papageorgiou, Effie
  last_name: Papageorgiou
citation:
  ama: Bruno T, Papageorgiou E. Blow-up exponents and a semilinear elliptic equation
    for the fractional Laplacian on hyperbolic spaces. <i>arXiv:250912349</i>. Published
    online 2025.
  apa: Bruno, T., &#38; Papageorgiou, E. (2025). Blow-up exponents and a semilinear
    elliptic equation for the fractional Laplacian on hyperbolic spaces. In <i>arXiv:2509.12349</i>.
  bibtex: '@article{Bruno_Papageorgiou_2025, title={Blow-up exponents and a semilinear
    elliptic equation for the fractional Laplacian on hyperbolic spaces}, journal={arXiv:2509.12349},
    author={Bruno, Tommaso and Papageorgiou, Effie}, year={2025} }'
  chicago: Bruno, Tommaso, and Effie Papageorgiou. “Blow-up Exponents and a Semilinear
    Elliptic Equation for the Fractional Laplacian on Hyperbolic Spaces.” <i>ArXiv:2509.12349</i>,
    2025.
  ieee: T. Bruno and E. Papageorgiou, “Blow-up exponents and a semilinear elliptic
    equation for the fractional Laplacian on hyperbolic spaces,” <i>arXiv:2509.12349</i>.
    2025.
  mla: Bruno, Tommaso, and Effie Papageorgiou. “Blow-up Exponents and a Semilinear
    Elliptic Equation for the Fractional Laplacian on Hyperbolic Spaces.” <i>ArXiv:2509.12349</i>,
    2025.
  short: T. Bruno, E. Papageorgiou, ArXiv:2509.12349 (2025).
date_created: 2026-02-19T11:42:22Z
date_updated: 2026-02-19T11:43:16Z
external_id:
  arxiv:
  - '2509.12349'
language:
- iso: eng
project:
- _id: '357'
  name: 'TRR 358: Ganzzahlige Strukturen in Geometrie und Darstellungstheorie'
publication: arXiv:2509.12349
status: public
title: Blow-up exponents and a semilinear elliptic equation for the fractional Laplacian
  on hyperbolic spaces
type: preprint
user_id: '100325'
year: '2025'
...
---
_id: '64266'
abstract:
- lang: eng
  text: We study the large-time asymptotic behavior of solutions to the discrete-time
    heat equation, i.e., caloric functions, on affine buildings, including those without
    transitive group actions. For each $p \in [1, \infty]$, we introduce a notion
    of a $p$-mass function and prove that caloric functions with initial data belonging
    to certain weighted-$\ell^1$ spaces or to the radial $\ell^1$ class, asymptotically
    decouple as the product of this mass function and the heat kernel. These results
    extend classical analogues from Euclidean spaces and symmetric spaces of non-compact
    type to the non-Archimedean setting, and remain valid even for exotic buildings
    beyond the Bruhat--Tits framework. We characterize the spatial concentration of
    heat kernels in $p$-norms and describe the geometry of associated critical regions.
    Our results highlight substantial differences in the asymptotic regimes depending
    on the value of $p$, and clarify the interplay between volume growth and heat
    diffusion.
author:
- first_name: Effie
  full_name: Papageorgiou, Effie
  last_name: Papageorgiou
- first_name: Bartosz
  full_name: Trojan, Bartosz
  last_name: Trojan
citation:
  ama: Papageorgiou E, Trojan B. Mass Functions and Asymptotic Behavior of Caloric
    Functions on Affine Buildings. <i>arXiv:250617042</i>. Published online 2025.
  apa: Papageorgiou, E., &#38; Trojan, B. (2025). Mass Functions and Asymptotic Behavior
    of Caloric Functions on Affine Buildings. In <i>arXiv:2506.17042</i>.
  bibtex: '@article{Papageorgiou_Trojan_2025, title={Mass Functions and Asymptotic
    Behavior of Caloric Functions on Affine Buildings}, journal={arXiv:2506.17042},
    author={Papageorgiou, Effie and Trojan, Bartosz}, year={2025} }'
  chicago: Papageorgiou, Effie, and Bartosz Trojan. “Mass Functions and Asymptotic
    Behavior of Caloric Functions on Affine Buildings.” <i>ArXiv:2506.17042</i>, 2025.
  ieee: E. Papageorgiou and B. Trojan, “Mass Functions and Asymptotic Behavior of
    Caloric Functions on Affine Buildings,” <i>arXiv:2506.17042</i>. 2025.
  mla: Papageorgiou, Effie, and Bartosz Trojan. “Mass Functions and Asymptotic Behavior
    of Caloric Functions on Affine Buildings.” <i>ArXiv:2506.17042</i>, 2025.
  short: E. Papageorgiou, B. Trojan, ArXiv:2506.17042 (2025).
date_created: 2026-02-19T11:41:25Z
date_updated: 2026-02-19T11:43:53Z
external_id:
  arxiv:
  - '2506.17042'
language:
- iso: eng
project:
- _id: '357'
  name: 'TRR 358: Ganzzahlige Strukturen in Geometrie und Darstellungstheorie'
publication: arXiv:2506.17042
status: public
title: Mass Functions and Asymptotic Behavior of Caloric Functions on Affine Buildings
type: preprint
user_id: '100325'
year: '2025'
...
---
_id: '56717'
abstract:
- lang: eng
  text: "We establish a multiresolution analysis on the space $\\text{Herm}(n)$ of\r\n$n\\times
    n$ complex Hermitian matrices which is adapted to invariance under\r\nconjugation
    by the unitary group $U(n).$ The orbits under this action are\r\nparametrized
    by the possible ordered spectra of Hermitian matrices, which\r\nconstitute a closed
    Weyl chamber of type $A_{n-1}$ in $\\mathbb R^n.$ The space\r\n$L^2(\\text{Herm}(n))^{U(n)}$
    of radial, i.e. $U(n)$-invariant $L^2$-functions\r\non $\\text{Herm}(n)$ is naturally
    identified with a certain weighted $L^2$-space\r\non this chamber.\r\n  The scale
    spaces of our multiresolution analysis are obtained by usual dyadic\r\ndilations
    as well as generalized translations of a scaling function, where the\r\ngeneralized
    translation is a hypergroup translation which respects the radial\r\ngeometry.
    We provide a concise criterion to characterize orthonormal wavelet\r\nbases and
    show that such bases always exist. They provide natural orthonormal\r\nbases of
    the space $L^2(\\text{Herm}(n))^{U(n)}.$\r\n  Furthermore, we show how to obtain
    radial scaling functions from classical\r\nscaling functions on $\\mathbb R^{n}$.
    Finally, generalizations related to the\r\nCartan decompositions for general compact
    Lie groups are indicated."
article_type: original
author:
- first_name: Lukas
  full_name: Langen, Lukas
  id: '73664'
  last_name: Langen
- first_name: Margit
  full_name: Rösler, Margit
  id: '37390'
  last_name: Rösler
citation:
  ama: Langen L, Rösler M. Multiresolution analysis on spectra of hermitian matrices.
    <i>Indagationes Mathematicae</i>. 2025;36(6):1671-1694.
  apa: Langen, L., &#38; Rösler, M. (2025). Multiresolution analysis on spectra of
    hermitian matrices. <i>Indagationes Mathematicae</i>, <i>36</i>(6), 1671–1694.
  bibtex: '@article{Langen_Rösler_2025, title={Multiresolution analysis on spectra
    of hermitian matrices}, volume={36}, number={6}, journal={Indagationes Mathematicae},
    publisher={Elsevier}, author={Langen, Lukas and Rösler, Margit}, year={2025},
    pages={1671–1694} }'
  chicago: 'Langen, Lukas, and Margit Rösler. “Multiresolution Analysis on Spectra
    of Hermitian Matrices.” <i>Indagationes Mathematicae</i> 36, no. 6 (2025): 1671–94.'
  ieee: L. Langen and M. Rösler, “Multiresolution analysis on spectra of hermitian
    matrices,” <i>Indagationes Mathematicae</i>, vol. 36, no. 6, pp. 1671–1694, 2025.
  mla: Langen, Lukas, and Margit Rösler. “Multiresolution Analysis on Spectra of Hermitian
    Matrices.” <i>Indagationes Mathematicae</i>, vol. 36, no. 6, Elsevier, 2025, pp.
    1671–94.
  short: L. Langen, M. Rösler, Indagationes Mathematicae 36 (2025) 1671–1694.
date_created: 2024-10-22T09:31:19Z
date_updated: 2026-02-19T14:16:43Z
ddc:
- '510'
department:
- _id: '555'
external_id:
  arxiv:
  - '2410.10364'
file:
- access_level: closed
  content_type: application/pdf
  creator: llangen
  date_created: 2026-02-19T14:14:39Z
  date_updated: 2026-02-19T14:14:39Z
  file_id: '64288'
  file_name: MSA_hermitsch_published.pdf
  file_size: 443262
  relation: main_file
  success: 1
file_date_updated: 2026-02-19T14:14:39Z
has_accepted_license: '1'
intvolume: '        36'
issue: '6'
language:
- iso: eng
main_file_link:
- url: https://doi.org/10.1016/j.indag.2025.03.009
page: 1671-1694
project:
- _id: '357'
  name: TRR 358 - Ganzzahlige Strukturen in Geometrie und Darstellungstheorie
publication: Indagationes Mathematicae
publication_status: published
publisher: Elsevier
related_material:
  link:
  - relation: research_paper
    url: https://arxiv.org/abs/2410.10364
status: public
title: Multiresolution analysis on spectra of hermitian matrices
type: journal_article
user_id: '73664'
volume: 36
year: '2025'
...